Hw2

1

Given {b1} = 0.5, the power of the test with 48 degrees of freedom, for a true slope of 1.7, and a null of 0 would be 0.84 then there would be a 84% change of rejecting the null given that the true β1 was 1.7.

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2

A

Found by hand the 90% CI for a stoping distance given a starting speed of 15mph is...

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B

read.csv("Stop.csv", header = TRUE) -> data

n <-length(data) 
X <-data$speed
Y <-data$dist

reg.stop <-lm(Y ~ X)

new.dat <-data.frame(X=15)
predict(reg.stop, newdata=new.dat, interval="confidence", level=0.90)

##        fit      lwr      upr
## 1 41.40704 37.74843 45.06564

ci <- predict(reg.stop, newdata=new.dat, interval="confidence", level=0.90) 

C

We can be 90% confident that the mean stopping distance, in ft, at 15mph is somewhere between (37.748435), (45.065638)

3

A

90% prediction interval for the stopping distance of a new driver whose speed is 15 mph.

Drawing.

B

predict(reg.stop, newdata=new.dat, interval="predict", level=0.90)

##        fit      lwr      upr
## 1 41.40704 15.35386 67.46022

ci <- predict(reg.stop, newdata=new.dat, interval="predict", level=0.90)

C

We can be 90% confident in predicting that a new stopping test conducted at 15mph would produce a stopping distance, in ft, somewhere between (15.3538569), (67.4602161)

4

A

A 90% prediction interval for the mean stopping distance of three new drivers each of whose speed is 15 mph.

Drawing.

B

ci.reg(reg.stop, new.dat, type = 'nm', alpha=0.10,m=3) -> threeRep

threeRep[3:4]

##   Lower.Band Upper.Band
## 1   26.07147    56.7426

C

The interval for the total stopping distance for all three tests would be between 78.2144069 and 170.2278121

5

A 95% confidence band for the simple linear regression line predicting stopping distance using speed.

ggplot(data, aes(x=speed, y=dist)) + 
  geom_point(color='#2980B9', size = 4) + 
  geom_smooth(method=lm, color='#2C3E50') +
  xlab("Spead (mph)") +
  ylab("Stopping Dist. (ft)") +
  theme_minimal()

## `geom_smooth()` using formula 'y ~ x'